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1. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
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On Improved Estimation of Population Mean using Qualitative
Auxiliary Information
SUBHASH KUMAR YADAV1*, A.A. ADEWARA2
1. Department of Mathematics and Statistics (A Centre of Excellence), Dr. RML Avadh University, Faizabad224001, U.P., INDIA
2. Department of Statistics, University of Ilorin, Ilorin. Kwara State, Nigeria
* Corresponding author: drskystats@gmail.com
Abstract
This paper deals with the estimation of population mean of the variable under study by improved ratio-product
type exponential estimator using qualitative auxiliary information. The expression for the bias and mean squared
error (MSE) of the proposed estimators has been derived to the first order of approximation. A comparative
approach has been adopted to study the efficiency of proposed and previous estimators. The present estimators
provide us significant improvement over previous estimators leading to the better perspective of application in
various applied areas. The numerical demonstration has been presented to elucidate the novelty of paper.
Keywords: Exponential estimator, auxiliary attribute, Proportion, bias, mean squared error, efficiency.
Mathematics Subject Classification 2010: 62D05
1. Introduction
The use of supplementary (auxiliary) information has been widely discussed in sampling theory. Auxiliary
variables are in use in survey sampling to obtain improved sampling designs and to achieve higher precision in
the estimates of some population parameters such as the mean or the variance of the variable under study. This
information may be used at both the stage of designing (leading for instance, to stratification, systematic or
probability proportional to size sampling designs) and estimation stage. It is well established that when the
auxiliary information is to be used at the estimation stage, the ratio, product and regression methods of
estimation are widely used in many situations.
The estimation of the population mean is a burning issue in sampling theory and many efforts have been
made to improve the precision of the estimates. In survey sampling literature, a great variety of techniques for
using auxiliary information by means of ratio, product and regression methods has been used. Particularly, in
the presence of multi-auxiliary variables, a wide variety of estimators have been proposed, following different
ideas, and linking together ratio, product or regression estimators, each one exploiting the variables one at a time.
The first attempt was made by Cochran (1940) to investigate the problem of estimation of population
mean when auxiliary variables are present and he proposed the usual ratio estimator of population mean. Robson
(1957) and Murthy (1964) worked out independently on usual product estimator of population mean. Olkin
(1958) also used auxiliary variables to estimate population mean of variable under study. He considered the
linear combination of ratio estimators based on each auxiliary variable separately making use of information
related to the supplementary characteristics having positive correlation with the variable under consideration.
Singh (1967a) dwelt upon a multivariate expression of Murthy’s (1964) product estimator. Further, the
multi-auxiliary variables through a linear combination of single difference estimators were attempted by Raj
(1965). In next bid of investigation Singh (1967b) extended the ratio-cum-product estimators to multisupplementary variables. An innovative idea of weighted sum of single ratio and product estimators leading to
multivariate was developed by Rao and Mudholkar (1967). Much versatile effort was made by John (1969) by
considering a general ratio-type Estimator that, in turn, presented n unified class of estimators obtaing various
particular estimators suggested by previous authors such as Olkin’s (1958) and Singh’s (1967a). Srivastava
(1971) dealt with a general ratio-type estimator unifying previously developed estimators by eminent authors
engaged in this area of investigation. Searls (1964) and Sisodia & Dwivedi (1981) used coefficient of variation
of study and auxiliary variables respectively to estimate population mean of study variable.
Srivenkataramana (1980) first proposed the dual to ratio estimator for estimating population mean.
Kadilar and Cingi (2004, 2005) analyzed combinations of regression type estimators in the case of two auxiliary
variables. In the same situation, Perri (2005) proposed some new estimators obtained from Singh’s (1965, 1967b)
ones. Singh and Tailor (2005), Tailor and Sharma (2009) worked on ratio-cum-product estimators. Sharma and
Tailor (2010) proposed a ratio-cum-dual to ratio estimator for the estimation of finite population mean of the
study variable y. In the series of improvement Das and Tripathi (1978), Srivastava and Jhajj (1980), Singh et.al
(1988), Prasad and Singh (1990, 1992), Naik and Gupta (1991), Ceccon and Diana (1996), Agarwal et al. (1997),
Upadhyay and Singh (1999, 2001), Abu-Dayyeh et al. (2003), Javid Shabbir (2006), Kadilar and Cingi (2006),
Singh et.al (2007), Singh et.al (2009), Muhammad Hanif et.al (2010), Yadav (2011), Pandey et.al (2011), Shukla
42

2. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
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et.al (2012), Onyeka (2012) etc have proposed many estimators utilizing auxiliary information. In the present
study, we suggest a new estimator for estimating population mean of the variable under study.
The use of auxiliary information may increase the precision of an estimator when study variable Y is
highly correlated with auxiliary variable X. When the variable under study Y is highly positively correlated
with the auxiliary variable X, then the ratio type estimators are used to estimate the population parameter and
product estimators are used when the variable under study Y is highly negatively correlated with the auxiliary
variable X for improved estimation of parameters of variable under study. However there are situations when
information on auxiliary variable is not available in quantitative form but in practice, the information regarding
the population proportion possessing certain attribute
ψ
is easily available (see Jhajj et.al. [7]), which is
highly associated with the study variable Y. For example (i) Y may be the use of drugs and ψ may be the
gender (ii) Y may be the production of a crop and ψ may be the particular variety. (iii) Y may be the amount
of milk produced and ψ a particular breed of cow. (iv) Y may be the yield of wheat crop and ψ a particular
variety of wheat etc. (see Shabbir and Gupta [25]).
Let there be N units in the population. Let
( yi ,ψ i ) , i = 1, 2, ….. , N be the corresponding observation
values of the ith unit of the population of the study variable Y and the auxiliary variable
Further we assume that
ψ
respectively.
ψ i = 1 and ψ i = 0, i = 1, 2, …. , N if it possesses a particular characteristic or does
N
n
A = ∑ψ i
i =1
not possess it. Let
a = ∑ψ i
i =1
and
sample respectively possessing the attributeψ
denote the total number of units in the population and
P=
. Let
A
a
p=
N and
n denote the proportion of units in the
population and sample respectively possessing the attributeψ . Let a simple random sample of size n from this
( yi ,ψ i ) , i = 1, 2,….., n.
population is taken without replacement having sample values
Naik and Gupta [15] defined the following ratio and product estimators of population mean when
the prior information of population proportion of units, possessing the same attributeψ is available as
P
t1 = y ( )
p
(1.1)
p
t2 = y ( )
P
(1.2)
t1 and t2 up to the first order of approximation are
2
2
MSE (t1 ) = f Y 2 [C y + C p (1 − 2 K p )]
The MSE of estimators
(1.3)
MSE (t 2 ) = f Y [C + C (1 + 2 K p )]
2
2
y
2
p
(1.4)
Singh et.al [27] defined the following ratio, product and ratio & product estimators respectively of
population mean using qualitative auxiliary information as
t3 = y exp(
t4 = y exp(
P− p
)
P+ p
(1.5)
p−P
)
p−P
(1.6)

P− p
p−P 
t5 = y α exp(
) + (1 − α ) exp(
)
P+ p
p−P 

(1.7)
t
t
Where α is a real constant to be determined such that the MSE of 5 is minimum. For α = 1 , 5 reduces to the
43

3. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
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and for α
= 0 , it reduces to the estimator t4 .
t t
t
The MSE of the estimators 3 , 4 and 5 up to the first order of approximation are respectively as
estimator
t3
2
2 1
MSE (t 3 ) = f Y 2 [C y + C p ( − K p )]
4
2
2 1
MSE (t 4 ) = f Y 2 [C y + C p ( + K p )]
4
1
2
2 1
MSE (t5 ) = f Y 2 [C y + C p ( + α 2 − α ) + 2 ρ pbC y C p ( − α )]
4
2
(1.8)
(1.9)
(1.10)
which is minimum for optimum value of α as
2K p + 1
α=
= α 0 ( say )
2
t5 is
2
2
MSEmin (t5 ) = f Y 2C y (1 − ρ pb ) = M (t5 ) opt
and the minimum MSE of
which is same as that of traditional linear regression estimator.
where
and
2
Sy
2
Sψ
Kp =
C = 2
Y2,
P ,
2
N
1
2
Sψ = ∑ (ψ i − P)
N i =1
,
S yψ
ρ pb =
S y Sψ
C =
2
y
2
p
Cy p
S yψ =
2
Cp
1
N
= ρ pb
N
∑ (Y
i
Cy
2
1 N
S = ∑ (Yi − Y )
Cp
N i =1
,
,
2
y
− Y )(ψ i − P)
i =1
f =
,
1 1
−
n N
is the point biserial correlation coefficient.
2. Suggested Estimators
Motivated by Prasad [19] and Gandge et al. [6], we propose
The exponential ratio type estimator as
ξ1 = ky exp(
P− p
)
P+ p
(2.1)
The exponential product type estimator as
ξ 2 = ky exp(
p−P
)
p−P
(2.2)
The exponential dual to ratio type estimator as
 p* − P 
ξ3 = y exp *
 p +P



( NP − nψ i )
ψ i* =
ψ i* = (1 + g ) P − gψ i , i = 1, 2, ......, N
( N − n)
where
or
n
g=
*
p = (1 + g ) P − gp where
N −n .
The exponential ratio and dual to ratio type estimator as
44
(2.3)
,
which
usually
gives

4. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

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 p * − P 
P− p
 + (1 − α ) exp *

 p + P 

P+ p


ξ 4 = y α exp


ξ
where α is a real constant to be determined such that the MSE of 4
t
ξ
the estimator 3 and for α = 0 , it reduces to the estimator 3 .
(2.4)
ξ
is minimum. For α = 1 , 4
reduces to
To obtain the bias and mean squared error (MSE) of the estimators, let
y = Y (1 + e0 )
2
2
E (e0 ) = f C y
p = P(1 + e1 ) such that E (ei ) = 0 , i = 0,1 and
2
E (e0e1 ) = f C yp = fρ pb C y C p
E (e12 ) = f C p
and
,
and
y and p , we have
From (2.1) by putting the values of
 P − P (1 + e1 ) 

 P + P (1 + e1 ) 
ξ1 = kY (1 + e0 ) exp 
 − e1 
= kY (1 + e0 ) exp 

 2 + e1 
− e 
ξ1 = kY (1 + e0 ) exp 1 
 2 
(2.5)
Expanding the right hand side of (2.5) and retaining terms up to second powers of e’s, and then subtracting
from both sides, we have

ξ1 − Y = kY 1 + e0 −

e1 e12 e0 e1 
+ −
 −Y
2 8
2 
Y
(2.6)
Taking expectations on both sides of (2.6), we get the bias of the estimator
approximation, as
ξ1
up to the first order of
2
Cp 1
B(ξ1 ) = k Y f
( − K p ) + Y ( k − 1)
2 4
(2.7)
Squaring both sides of equation (2.6) gives
2


e e2 e e 
e e2 e e 
(ξ1 − Y ) = k Y 1 + e0 − 1 + 1 − 0 1  + Y 2 − 2kY 2 1 + e0 − 1 + 1 − 0 1 
2 8
2 
2 8
2 


ξ
and now taking expectation, we get the MSE of the estimator 1 , to the first order of approximation as
2
2
2
MSE (ξ 1 ) = f Y [ k fC + ( 2k − k ) f (
2
The minimum of
2
2
y
2
2
Cp
4
− ρ pb C y C p ) + ( k − 1) 2 ]
(2.8)
MSE(ξ1 ) is obtained for the optimal value of k, which is
k opt =
A1
2 B1
(2.9)
where
A1 = f (
2
Cp
4
− ρ pb C y C p ) + 2
2
B1 = fC y + 2 f (
and
Thus the minimum MSE of the estimator
ξ1
is obtained as
45
2
Cp
4
− ρ pb C y C p ) + 1

5. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
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
A2 
MSE min (ξ1 ) = Y 2 1 − 1 
 4 B1 
(2.10)
Following the same procedure as above, we get the bias and MSE of the estimator
approximation, as
ξ2
up to the first order of
2
Cp 1
B(ξ 2 ) = k Y f
( + K p ) + Y ( k − 1)
2 4
2
MSE (ξ 2 ) = f Y 2 [ k 2 fC y + ( 2k 2 − k ) f (
C
2
p
4
(2.11)
+ ρ pb C y C p ) + ( k − 1) 2 ]
(2.12)
The minimum MSE of the estimator ξ 2 is obtained for optimal value of k as

A2 
MSE min (ξ 2 ) = Y 2 1 − 2 
 4 B2 
(2.13)
where the optimal value of k for the estimator ξ 2 is
k opt =
A2 = f (
2
Cp
A2
2 B2
+ ρ pb C y C p ) + 2
B2 = fC + 2 f (
2
y
4
where
Expressing (2.3) in terms of e’s, we have
2
Cp
and
4
+ ρ pbC y C p ) + 1
 ge1 

 2 
ξ 3 = Y (1 + e0 ) exp
(2.14)
Expanding the right hand side of (2.14) and retaining terms up to second powers of
e’s, and then subtracting
Y from both sides, we have

ge1 g 2 e12 ge0 e1 
ξ 3 − Y = Y 1 + e0 +
+
+
 −Y
2
8
2 

(2.15)
ξ
Taking expectations on both sides of (2.15), we get the bias of the estimator 3 up to the first order of
approximation, as
2
Cp g
B(ξ 3 ) = Y fg
( + Kp)
2 4
(2.16)
From equation (2.15), we have
(ξ 3 − Y ) ≅ Y (e0 +
ge1
)
2
(2.17)
Squaring both sides of (2.17) and then taking expectations, we get MSE of the estimator
of approximation as
2
2
MSE (ξ 3 ) = f Y 2 [C y + gC p (
g
+ K p )]
4
ξ 3 , up to the first order
(2.18)
Expressing (2.4) in terms of e’s, we have

 − e1 
 ge 
 + (1 − α ) exp 1 
 2 
 2 
ξ 4 = Y (1 + e0 ) α exp

(2.19)
Expanding the right hand side of (2.19) and retaining terms up to second powers of e’s, and then subtracting
from both sides, we have
46
Y

6. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

ξ 4 − Y = Y 1 + e0 + α1

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e1
e2
ee 
+ α 2 1 + α1 0 1  − Y
2
8
2 
(2.20)
α = [g − (1 + g )α ] and α 2 = [ g 2 + (1 − g 2 )α ] .
Where 1
Taking expectations on both sides of (2.20), we get the bias of the estimator
approximation, as
ξ4
up to the first order of
2
 C p α1

+ ρ pbC y C p 
B(ξ 4 ) = fY α 2
2
 8



(2.21)
From equation (2.20), we have
(ξ 4 − Y ) ≅ Y (e0 + α1
e1
)
2
(2.22)
Squaring both sides of equation (2.22) gives
 2

e2
(ξ 4 − Y ) 2 = Y 2 e0 + α12 1 + α1e0e1 
4


and now taking expectation, we get the MSE of the estimator
2
MSE (ξ 4 ) = f Y 2 [C y + α12
2
Cp
4
(2.23)
ξ 4 , to the first order of approximation as
+ α1ρ pbC y C p ]
(2.24)
which is minimum for optimum value of α as
2K p + g
α=
= α opt (say)
1+ g
ξ
and the minimum MSE of 4 is
2
2
MSEmin (ξ 4 ) = f Y 2C y (1 − ρ pb ) = M (ξ 4 ) opt
(2.25)
which is same as that of traditional linear regression estimator and also equal to
3. Efficiency comparisons
Following are the conditions for which the proposed estimator
We know that the variance of the sample mean
ξ4
(t5 ) opt
.
is better than the
t and ξ estimators.
y is
2
V ( y ) = f Y 2C y
(3.1)
ξ
To compare the efficiency of the proposed estimator 4
with the existing and proposed estimators, from (3.1)
and (1.3), (1.4), (1.8), (1.9), (2.10), (2.13) and (2.18), we have
2
V ( y ) − M (ξ 4 ) opt = ρ pb ≥ 0
(3.2)
MSE (t1 ) − M (ξ 4 ) opt = (C p − ρ pbC y ) ≥ 0
(3.3)
MSE (t2 ) − M (ξ 4 ) opt = (C p + ρ pbC y ) ≥ 0
(3.4)
2
2
MSE (t3 ) − M (ξ 4 ) opt = (
MSE (t4 ) − M (ξ 4 ) opt = (
Cp
2
Cp
2
− ρ pbC y ) 2 ≥ 0
(3.5)
+ ρ pbC y ) 2 ≥ 0
(3.6)
47

7. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
M (ξ 4 ) opt < MSE (ξ1 )
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f (1 − ρ 2 ) < (1 −
if
f (1 − ρ 2 ) < (1 −
M (ξ 4 ) opt < MSE (ξ 2 )
if
M (ξ 4 ) opt < MSE (ξ3 )
if
A12
)
4 B1
(3.7)
2
2
A
)
4 B2
(3.8)
gC p + 8 ρ pbC y < 0
(3.9)
4. Empirical Study
To analyze the performance of various estimators of population mean
following two data sets
Data 1. [Source: Sukhatme & Sukhatme [33], page 305]
Y of study variable y, we considered the
Y = Area (in acres) under wheat crop in the circles and ψ = A circle consisting more than five villages.
N = 89 , n = 23 , Y = 3.360 , P = 0.1236 , ρ pb = 0.766 , C y = 0.60400 , C p = 2.19012
Data 2. [Source: Mukhopadhyay [12], page 44]
Y = Household size and ψ = A household that availed an agricultural loan from a bank.
N = 25 , n = 7 , Y = 9.44 , P = 0.400 , ρ pb = - 0.387 , C y = 0.17028 , C p = 1.27478
y
The percent relative efficiency (PRE) of the estimators , proposed and mentioned existing estimators and
(ξ 4 ) opt
y usual unbiased estimator have been computed and given in table1.
with respect to
y
Table1: PRE of various estimators with respect to
y
PRE of . , with respect to
Estimator
Population I
100.00
y
t1
t2
t3
Population II
100.00
11.63
1.59
5.07
1.95
66.25
5.58
t4
14.16
8.28
ξ1
ξ2
ξ3
67.84
6.23
16.58
8.89
42.11
50.17
241.99
117.62
(ξ 4 ) opt (t5 ) opt
=
5. Conclusion
From Table1 we see that the proposed
estimator
ξ4
under
optimum condition
performs better than the
t
y
t
t
t
usual sample mean estimator , Naik and Gupta estimators ( 1 and 2 ), Singh et.al estimators ( 3 and 4 ),
ξ
ξ ξ
proposed estimators ( 1 , 2 and 3 ). The ξ estimators are better than the corresponding t estimators and under
optimum conditions
ξ4
and
t5
both are equally efficient.
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8. Mathematical Theory and Modeling
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